Month: May 2013

Rebecca Christainsen had the highest score of any student on our Do You Know Blue machine learning activity. Yesterday was her last day of school at Terman Middle School in Palo Alto, CA, so Evan Weinberg, Dave Major, and I sent her math class a pizza party in her honor.

Because we’re keeping the activity available for you and your students to use as they study inequalities, we aren’t going to go into much depth on all the different rules contestants used. But I asked Rebecca how she came to her final, game-winning rule, and she told all:

My teacher first showed me the website, and I decided to try it out. My first attempt scored me only around 18%, but since hardly anyone had tried it out yet, I was ranked 33rd. After that, I was encouraged to try more equations, and suddenly thought of all the different types of equations that I could use, and moved to squared terms. One of the first equations that I came up with was b2>r2+g2. I simply used trial and error to come up with new equations, and I recorded each equation that I used and the percentage. I combined different equations together, and a few different combinations even had the same percentage.

I was in Australia this last week, working with some teachers at MYSA on You Pour, I Choose. It’s a task that asks which of two glasses has more soda and involves, among other skills, a fairly straightforward application of volume.

A teacher in the workshop called me over. “I’m not a math teacher,” she told me, and then pointed to the person next to her who had calculated the formula for volume of a cylinder.

“But that seems like more work than you need to do,” she said. “We don’t care about the exact amount. We care which one has more. With both glasses, we multiply by pi and square the radius. So all you really need to do is multiply the radius by the height for both glasses and compare the result. That’ll tell you which one has more.”

This was a rather stunning suggestion, made all the more impressive by the fact that this woman doesn’t immerse herself in numbers and variables for a living like the rest of us.

I have two questions:

Is she right? She’s certainly right in this case. Both the volume formula and her shortcut indicate the left glass has more soda.

As her teacher, what do you do next?

I’ll update this post tomorrow.

2013 May 29. I knew just telling her, “That’s wrong.” would be unsatisfying because, explicitly, she said, “Am I wrong?” but, implicitly, she was saying, “If I’m wrong, then make me believe it.” I knew the current problem wasn’t helping me out because her shortcut actually worked.

“Whoa,” I said. “Does that work? If that works that’s going to save us a lot of time going forward. Let me bring your idea to the group and see what everybody thinks.”

In the meantime I stewed over a counterexample. It took me more than a minute to think of one because a) I was kind of adrenalized by the whole exchange, and b) I don’t do this on a daily basis anymore so my counterexample-finding muscle has become doughy and underused.

I posed her shortcut to the group and said, “What do you think? Does this work?” I gave them time to think and debate about it. Someone came back and said, “No, it doesn’t work. Imagine two cylinders with different heights and a radius of one.”

Awesome, right? This particular counterexample doesn’t disprove the rule. The square of one is also one so her rule works here also.

Eventually someone suggested two examples where the product of the radius and height were the same but where the radius and the height were different in each cylinder. The shortcut says they should have the volume. The formula for volume says they’re different.

Final note: students are often asked to prove conjectures that are either a) totally obvious (“the sum of two even numbers is even” in high school) or b) totally abstract (“prove the slopes of two perpendicular lines are negative reciprocals” in middle school). It’s rare to find a conjecture that is both easily understood by the class and not obviously correct or incorrect. I’m filing this one away.

If you and I have had a conversation about math education in the last month, it’s likely I’ve taken you by the collar, stared straight at you, and said, “Can I tell you about the math lesson that has me most excited right now?”

There was probably some spittle involved.

Evan Weinberg posted “(Students) Thinking Like Computer Scientists” a month ago and the lesson idea haunted me since. It realizes the promise of digital, networked math curricula as well as anything else I can point to. If math textbooks have a digital future, you’re looking at a piece of it in Evan’s post.

Evan’s idea basically demanded a full-scale Internetization so I spent the next month conspiring with Evan and Dave Major to put the lesson online where anybody could use it.

It’s so easy to start. While most modeling lessons begin by throwing information and formulas and dense blocks of text at students, Evan’s task begins with the concise, enticing, intuitive question “Is this blue?” That’s the power of a digital math curriculum. The abstraction can just wait a minute. We’ll eventually arrive at all those equations and tables and data but we don’t have to start with them.

Students embed their own data in the problem. By judging ten colors at the start of the task, students are supplying the data they’ll try to model later. That’s fun.

It’s a bridge from math to computer science. Students get a chance to write algorithms in a language understood by both mathematicians and the computer scientists. It’s analogous to the Netflix Prize for grown-up computer scientists.

It’s scaffolded. I won’t say we got the scaffolds exactly right, but we asked students to try two tasks in between voting on “blueness” and constructing a rule.

They guess, based on RGB values, if a color will be blue. This was instructive for me. It was obvious to me that a big number for blue and and little numbers for red and green would result in a blue color. I learned some other, more subtle combinations on this particular scaffold.

This is the modeling cycle. Modeling is often a cycle. You take the world, turn it into math, then you check the math against the world. In that validation step, if the world disagrees with your model, you cycle back and formulate a new model.

My three-act tasks rarely invoke the cycle, in contrast to Evan’s task. You model once, you see the answer, and then you discuss sources of error. But Evan’s activity requires the full cycle. You submit your first rule and it matches only 40% of the test data, so you cycle back, peer harder at the data, make a sharper observation, and then try a new model.

The contest is running for another five days. The top-ranked student, Rebecca Christainsen, has a rule that correctly predicts the blueness of 2,309 out of 2,594 colors for an overall accuracy of 89%. That’s awesome but not untouchable. Get on it. Get your students on it.

On April 19, 2013, the third day of NCTM’s annual meeting in Denver, Uri Treisman gave a forty-minute address on equity that Zal Usiskin, director of the University of Chicago’s School Mathematics Project, called the greatest talk he’d ever heard at the conference in any year. Stanford math professor Keith Devlin would later call it our “I have a dream” speech. At least one participant left in tears.

I’ve personally seen it three times. I got the video feed from NCTM and the slides from Treisman. I then spent some time stitching the two together, resulting in this video. His message is important enough that I’d like to use whatever technical skills I have, whatever time I have, whatever soapbox I can stand on, to help spread it.

If you’re interested in equity, you should watch it.
If you’re interested in teacher evaluation, you should watch it.
If you’re interested in school reform, you should watch it.
If you’re interested in charter schools, you should watch it.
If you’re interested in understanding which student outcomes teachers can control and which they can’t, you should watch it.
If you’re interested in the trajectory of math education in the era of the Common Core State Standards, you should watch it.

If none of those conditions apply to you, well, I can’t imagine the series of misclicks that brought you to my blog. Watch it.

Here’s a fair enough summary from Treisman himself:

There are two factors that shape inequality in this country and educational achievement inequality. The big one is poverty. But a really big one is opportunity to learn. As citizens, we need to work on poverty and income inequality or our democracy is threatened. As mathematics educators … we need to work on opportunity to learn. It cannot be that the accident of where a child lives or the particulars of their birth determine their mathematics education.

That was his destination and the talk took only three stops along the way:

What did education reform groups like Achieve, the Gates Foundation, et al, recommend in their “Benchmarking for Success” document in 2008?

How does TIMSS and NAEP data contradict or clarify those recommendations?

What should we actually do about equity, as teachers and citizens, if those recommendations prove unfounded?

Highly Quotable

[A]s math people we know that if we’re going to work on a problem, we have to formulate it clearly. And as math people are wont, we need to swaddle ourselves in the numbers and the data because that’s what gives math people direction, strength, and courage.

Let’s look at “Benchmarking for Success” and see its analysis of the problem. Then let’s look at the data and see how it actually lines up with what we know today. And then let’s see where we need to go to really enact the vision of NCTM for equity.

So the notion was: “Let’s focus on teachers as the central driver of reform and rethink how we evaluate teachers.” They had the view that teachers were the single most important in-school factor in student achievement. And math people know that was just an artifact of the way they modeled the problem.

About one half of students who go from high school to college are referred to remediation and mostly developmental math. Fewer than a quarter of those students will ever get a credential. Those students are more likely to end up with debt than a credential. [..] Those remedial programs are burial grounds for the aspirations of students. And it’s mostly math that’s the key trigger. 35,000 students in California two years ago repeated a developmental course for the fifth or greater number of times. So no one can say those students don’t have persistence.

So states – where you go to school – are a profound influence on what you actually get to know.

Low income student scores in Texas were the top in the country in 2011. It’s really good for Texas to be the top of the country. Because whenever Texas does something well, everyone else is positive that they can do better. When Massachusetts is at the top, people go, “Ah, it’s just Massachusetts.”

Again, two and a half years difference in opportunity depending on where you happen to go to school. This is something that, as a math teaching profession, we can influence. Poverty is something we need to work on as citizens. Opportunity to learn is something we need to work on as math educators. That’s a core message for this talk.

So you would think that charters would fix this. Almost all the charters in Texas produced 0% of students who are college-ready. There are a few of them – one KIPP, one YES Prep, one IDEA, one Harmony – that are pretty good. Most of them are well below the public schools. So this theory of Achieve, NGA, CCSSO, Race to the Top, that charters were the answer? Not so clear when you actually climb into the numbers. The reverse looks true.

When you visit most math classrooms it’s like you’re in a Kafkaesque universe of these degraded social worlds where children are filling in bubbles rather than connecting the dots. It’s driven by a compliance mentality on tests that are neither worthy of our children nor worthy of the discipline they purport to reflect. That is the reality. That’s something that we as math educators can control.

What this shows is that the current theory about school improvement – that charters, Common Core, value-added measures of teaching are going to solve the problem – is profoundly wrong. That doesn’t mean we can’t use the Common Core powerfully to reboot our systems but it’s not the solution to the basic problems of schooling.

Guess what? Poverty really sucks. It’s incredibly hard. All the lifespan studies going back to the 1920s show that poverty and youth is a very hard force. We need to build fault-tolerant schools and systems if we’re actually going to address equity.

Just think about it. The great majority of our children finish our schools positive that there’s a whole list of things they’re not. They come out of schooling believing they’re not mathematical, they’re not artistic, they’re not philosophical, they’re not athletic. And these self-imposed beliefs undermine your sense of personal freedom, the font from which all freedoms come.

You have to remember that when the Common Core was created, they didn’t come to NCTM. They got David Coleman to write it and he brought his friend Jason Zimba to do the math. They did not come to NCTM. It’s time for us now – the professional societies – to talk about what standards should be and how to reshape the Common Core so that it reflects our best practice knowledge of schooling. Hard message, but a necessary message.

What is the determinant of whether you have a high-skill job in the US? Overwhelmingly it’s mathematics. It’s the single biggest factor in upward social and economic mobility. It’s our beloved subject. It would be wonderful if it were music instead of math. Think how great the country would be if everyone were striving to learn to play an instrument instead of factor quadratic equations but the fact is it is our discipline that is the primary determinant.

A beautiful speech by Uri and a great contribution by Dan to put the slides with the video.

However, the quote about how the Common Core was written garbles the history badly. Here is a summary of the process.

CCSSO and NGA appointed a work team of about 50 people—educators, mathematicians, teachers, policy people—and asked me to lead it. They also appointed a writing team of 3 people—Phil Daro, myself, and Jason Zimba—to draft the standards. We solicited progressions documents from selected individuals or groups in the work team. The three lead writers produced a first draft based on these progressions. There was also a feedback group of about 20 people. You can find the pdf with the names of all these people by googling “Common Core State Standards Work Team”.

Minor iterations of the standards were circulated to the working team for comment and critique, of which there was an abundance. Major drafts (about 3 or 4) were circulated to the feedback group and the 48 participating states, which also produced a huge amount of commentary. Finally, a public comment period starting in March 2010 elicited about 10,000 comments, of which we looked at every single actionable comment.

There were also numerous organization reviews, including one by NCTM. I spent a weekend in DC with a team from NCTM listening to their concerns, which resulted in significant changes to the standards. Jason and I also spent a weekend with teachers from AFT, one team for each grade band, who gave us detailed feedback that also resulted in changes to the document.